• Up
• Contents
• Index
• Search

Assume that G is a finite group of exponent m with k conjugacy classes of elements. The operators discussed here are concerned with the ring of class functions on G, defined to be the ring of complex-valued functions on G that are constant on conjugacy classes. This ring is made into a C-algebra by identifying c∈C with the constant function that is c everywhere. In fact we will restrict ourselves to functions with values that are elements of cyclotomic fields.

Elements of the ring, that is, objects of type AlgChtrElt, are represented by the k values (elements of some cyclotomic field Q(ζ_n)) on the classes. The numbering of those elements matches the numbering of the classes as returned by Classes applied to the underlying group G: Thus X[i] is the value of the character X on the i-th class, i.e. Classes(G)[i].

• Creation Functions
  • Structure Creation
    • ClassFunctionSpace(G) : Grp -> AlgChtr
    • ClassFunctionSpace(Q) : SeqEnum -> AlgChtr
  • Element Creation
    • elt< R | a₁, ..., a_k :parameters> : AlgChtr, FldCycElt, ..., FldCycElt -> AlgChtrElt
    • R ! a : AlgChtr, RngIntElt -> AlgChtrElt
    • Id(R) : AlgChtr -> AlgChtrElt
    • Zero(R) : AlgChtr -> AlgChtrElt
• The Table of Irreducible Characters
  • KnownIrreducibles(R) : AlgChtr -> SeqEnum
  • CharacterTable(G :parameters) : Grp -> SeqEnum
  • CharacterTableDS(G :parameters) : Grp -> SeqEnum, SeqEnum
  • Basis(R) : AlgChtr -> SeqEnum
  • CharacterTableConlon(G) : Grp -> SeqEnum
  • LinearCharacters(G): Grp -> SeqEnum
  • CharacterDegrees(G): GrpPerm -> SeqEnum
  • CharacterDegrees(G, z, p): GrpPC, GrpPCElt, RngIntElt -> SeqEnum
  • CharacterDegreesPGroup(G): GrpPC -> SeqEnum
  • RationalCharacterTable(G): GrpFin -> SeqEnum
  • SaveCharacterTable(T, F): SeqEnum[AlgChtrElt], MonStgElt ->
  • Example Chtr_CharacterTable (H98E1)
  • Example Chtr_CharacterTable2 (H98E2)
• Character Ring Operations
  • Related Structures
    • NumberOfClasses(R) : AlgChtr -> RngIntElt
    • ClassesData(R) : AlgChtr -> SeqEnum[Tup]
    • Group(R) : AlgChtr -> Grp
    • PowerMap(R) : AlgChtr -> Map
    • Kernel(x) : AlgChtrElt -> Grp
    • Centre(x) : AlgChtrElt -> Grp
    • CoefficientField(x) : AlgChtrElt -> Rng
    • CharacterField(x) : AlgChtrElt -> Rng
    • DegreeOfCharacterField(x) : AlgChtrElt -> RngIntElt
• Element Operations
  • Arithmetic
  • Predicates and Booleans
    • x in y : AlgChtrElt, AlgChtrElt -> BoolElt
    • x notin y : AlgChtrElt, AlgChtrElt -> BoolElt
    • IsCharacter(x) : AlgChtrElt -> BoolElt
    • IsGeneralizedCharacter(x) : AlgChtrElt -> BoolElt
    • IsIrreducible(x) : AlgChtrElt -> BoolElt
    • IsLinear(x) : AlgChtrElt -> BoolElt
    • IsFaithful(x) : AlgChtrElt -> BoolElt
    • IsReal(x) : AlgChtrElt -> BoolElt
    • IsPermutationCharacter(c) : AlgChtrElt -> BoolElt
    • Example Chtr_grp-ispermutationcharacter (H98E3)
  • Accessing Class Functions
    • T[i] : TabChtr, RngIntElt -> AlgChtrElt
    • T[i][j] : TabChtr, RngIntElt, RngIntElt -> FldCycElt
    • # T : SeqEnum -> RngIntElt
    • x(g) : AlgChtrElt, GrpElt -> FldCycElt
    • x[i] : AlgChtrElt, RngIntElt -> FldCycElt
    • # x : AlgChtrElt -> RngIntElt
  • Conjugation of Class Functions
    • x ^ g : AlgChtrElt, GrpElt -> AlgChtrElt
    • x ^ H : AlgChtrElt, Grp -> { AlgChtrElt }
    • GaloisConjugate(x, j) : AlgChtrElt, RngIntElt -> AlgChtrElt
    • GaloisOrbit(x) : AlgChtrElt -> { AlgChtrElt }
    • IsConjugate(x, y) : AlgChtrElt, AlgChtrElt -> BoolElt, RngIntElt
    • ClassPowerCharacter(x, j) : AlgChtrElt, RngIntElt -> AlgChtrElt
  • Functions Returning a Scalar
    • Degree(x) : AlgChtrElt -> RngIntElt
    • InnerProduct(x, y) : AlgChtrElt, AlgChtrElt -> FldCycElt
    • Order(x) : AlgChtrElt -> RngIntElt
    • Norm(x) : AlgChtrElt -> FldCycElt
    • Schur(x, k) : AlgChtrElt, RngIntElt -> FldCycElt
    • StructureConstant(G, i, j, k) : Grp, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
• The Schur Index
  • SchurIndex(x) : AlgChtrElt -> RngIntElt
  • SchurIndices(x) : AlgChtrElt -> SeqEnum
  • Example Chtr_SchurIndex (H98E4)
  • Example Chtr_recipe-for-schur-index (H98E5)
  • SchurIndexGroup(n: parameters) : RngIntElt -> GrpPC
  • CharacterWithSchurIndex(n: parameters) : RngIntElt -> AlgChtrElt. GrpPC
  • Attribute
    • AssertAttribute(x, "IsCharacter", b) : AlgChtrElt, MonStgElt, BoolElt ->
• Constructions for Characters
  • Induction, Restriction and Lifting
    • Induction(x, G) : AlgChtrElt, Grp -> AlgChtrElt
    • LiftCharacter(c, f, G) : AlgChtrElt, Map, Grp -> AlgChtrElt
    • LiftCharacters(T, f, G) : [AlgChtrElt], Map, Grp -> AlgChtrElt
    • Restriction(x, H) : AlgChtrElt, Grp -> AlgChtrElt
    • SubgroupFusion(G, H) : Grp, Grp -> SeqEnum[RngIntElt]
  • Symmetrization
    • Symmetrization(chi, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
    • OrthogonalSymmetrization(chi, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
    • SymplecticSymmetrization(chi, p) : AlgChtrElt, [ RngIntElt ] -> AlgChtrElt
    • Example Chtr_tensprod-check (H98E6)
    • Example Chtr_symp-and-orthog (H98E7)
    • IsOrthogonalCharacter(chi) : AlgChtrElt -> BoolElt
  • Permutation Character
    • PermutationCharacter(G) : GrpPerm -> AlgChtrElt
    • PermutationCharacter(G, H) : Grp, Grp -> AlgChtrElt
  • Composition and Decomposition
    • Composition(T, q) : [ AlgChtrElt ], [RngElt] -> AlgChtrElt
    • Decomposition(T, y) : [AlgChtrElt], AlgChtrElt -> [ FldCycElt ], AlgChtrElt
  • Finding Irreducibles
    • RemoveIrreducibles(I, C) : [ AlgChtrElt ], [ AlgChtrElt ] -> [ AlgChtrElt ], [ AlgChtrElt ]
    • ReduceCharacters(I, C) : [ AlgChtrElt ], [ AlgChtrElt ] -> [ AlgChtrElt ], [ AlgChtrElt ]
    • Example Chtr_A5 (H98E8)
• Brauer Characters
  • Structure Creation
    • CharacterRing(G, p) : Grp, RngIntElt -> AlgChtr
    • CharacterRing(R, p) : AlgChtr, RngIntElt -> AlgChtr
    • BrauerCharacteristic(R) : AlgChtr -> RngIntElt
  • Element Creation
    • R ! [ a₁, ..., a_k ] : AlgChtr, SeqEnum -> AlgChtrElt
    • Id(R) : AlgChtr -> AlgChtrElt
    • Zero(R) : AlgChtr -> AlgChtrElt
    • BrauerCharacter(x, p) : AlgChtrElt, RngIntElt -> AlgChtrElt
    • BrauerCharacter(M) : ModGrp -> AlgChtrElt
    • BrauerCharacterTable(G, p) : Chtr -> SeqEnum
  • Arithmetic
  • Induction, Restriction and Lifting
    • Induction(x, G) : AlgChtrElt, Grp -> AlgChtrElt
    • LiftCharacter(c, f, G) : AlgChtrElt, Map, Grp -> AlgChtrElt
    • LiftCharacters(T, f, G) : [AlgChtrElt], Map, Grp -> AlgChtrElt
    • Restriction(x, H) : AlgChtrElt, Grp -> AlgChtrElt
    • Blocks(T, p) : SeqEnum[AlgChtrElt], RngIntElt -> SeqEnum, SeqEnum
    • DefectGroup(T, b, p) : SeqEnum[AlgChtrElt], SetEnum[RngIntElt], RngIntElt -> Grp
    • DefectGroup(x, p) : AlgChtrElt, RngIntElt -> Grp
    • Example Chtr_brauer (H98E9)
• Database of Character Tables
  • CharacterTable(N) : MonStgElt -> SeqEnum[AlgChtrElt]
  • HasCharacterTable(A) : GrpAtlas -> BoolElt
  • CharacterTableNames() : -> SetIndx[MonStElt]
  • Example Chtr_CharacterTableData (H98E10)
  • Example Chtr_Lyons-schur-index (H98E11)
• Bibliography

• Up
• Contents
• Index
• Search